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His procedure is \ represented by the Greek letter mu (\[Micro]), and is called the ", ButtonBox["M\[ODoubleDot]bius function", ButtonData:>{ URL[ "http://mathworld.wolfram.com/MoebiusFunction.html"], None}, ButtonStyle->"Hyperlink"], " in his honor." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Table[MoebiusMu[n], {n, 1, 100}]\)], "Input"], Cell[BoxData[ \({1, \(-1\), \(-1\), 0, \(-1\), 1, \(-1\), 0, 0, 1, \(-1\), 0, \(-1\), 1, 1, 0, \(-1\), 0, \(-1\), 0, 1, 1, \(-1\), 0, 0, 1, 0, 0, \(-1\), \(-1\), \(-1\), 0, 1, 1, 1, 0, \(-1\), 1, 1, 0, \(-1\), \(-1\), \(-1\), 0, 0, 1, \(-1\), 0, 0, 0, 1, 0, \(-1\), 0, 1, 0, 1, 1, \(-1\), 0, \(-1\), 1, 0, 0, 1, \(-1\), \(-1\), 0, 1, \(-1\), \(-1\), 0, \(-1\), 1, 0, 0, 1, \(-1\), \(-1\), 0, 0, 1, \(-1\), 0, 1, 1, 1, 0, \(-1\), 0, 1, 0, 1, 1, 1, 0, \(-1\), 0, 0, 0}\)], "Output"] }, Open ]], Cell["\<\ In the \"zero\" bin, M\[ODoubleDot]bius placed multiples of square numbers \ (other than 1).\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(ZeroBin\ = \ Flatten[Position[Table[MoebiusMu[n], {n, 1, 100}], 0]]\)], "Input"], Cell[BoxData[ \({4, 8, 9, 12, 16, 18, 20, 24, 25, 27, 28, 32, 36, 40, 44, 45, 48, 49, 50, 52, 54, 56, 60, 63, 64, 68, 72, 75, 76, 80, 81, 84, 88, 90, 92, 96, 98, 99, 100}\)], "Output"] }, Open ]], Cell["Five square-multiples in a row.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Flatten[ Position[Partition[Table[MoebiusMu[n], {n, 1, 10000}], 5, 1], {0, 0, 0, 0, 0}]]\)], "Input"], Cell[BoxData[ \({844, 1680, 2888, 3624, 5046}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(MatrixForm[FactorInteger[844 + Range[0, 4]]]\)], "Input"], Cell[BoxData[ InterpretationBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\({{2, 2}, {211, 1}}\)}, {\({{5, 1}, {13, 2}}\)}, {\({{2, 1}, {3, 2}, {47, 1}}\)}, {\({{7, 1}, {11, 2}}\)}, {\({{2, 4}, {53, 1}}\)} }], "\[NoBreak]", ")"}], MatrixForm[ {{{2, 2}, {211, 1}}, {{5, 1}, {13, 2}}, {{2, 1}, {3, 2}, { 47, 1}}, {{7, 1}, {11, 2}}, {{2, 4}, {53, 1}}}]]], "Output"] }, Open ]], Cell["\<\ Differences between the actual and predicted density of square-divisible \ numbers.\ \>", "Text"], Cell[BoxData[ \(\(ListPlot[ FoldList[Plus, 0, Table[If[MoebiusMu[n] \[Equal] 0, 1, 0], {n, 1, 100000 - 1}]] - Table[n\ \((1 - 6/Pi^2)\), {n, 1, 100000}], PlotJoined \[Rule] True, AspectRatio \[Rule] 1/7];\)\)], "Input"], Cell["\<\ In the \"negative one\" bin, M\[ODoubleDot]bius placed any number that \ factored into an odd number of distinct primes.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(NegativeOneBin\ = \ Flatten[Position[Table[MoebiusMu[n], {n, 1, 100}], \(-1\)]]\)], "Input"], Cell[BoxData[ \({2, 3, 5, 7, 11, 13, 17, 19, 23, 29, 30, 31, 37, 41, 42, 43, 47, 53, 59, 61, 66, 67, 70, 71, 73, 78, 79, 83, 89, 97}\)], "Output"] }, Open ]], Cell["\<\ The number 665499549999999999 was found by using FromDigits and \ IntegerDigits, and selecting the numbers that satisfied MoebiusMu[n]=-1. \ When I noticed that some numbers could end in arbitrarily long strings of \ nines, I tried to find the best one using only 4 different digits.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Table[ MoebiusMu[FromDigits[Take[IntegerDigits[665499549999999999], n]]], {n, 2, 18}]\)], "Input"], Cell[BoxData[ \({\(-1\), \(-1\), \(-1\), \(-1\), \(-1\), \(-1\), \(-1\), \(-1\), \ \(-1\), \(-1\), \(-1\), \(-1\), \(-1\), \(-1\), \(-1\), \(-1\), \(-1\)}\)], \ "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(MatrixForm[ Table[Map[First, FactorInteger[ FromDigits[Take[IntegerDigits[665499549999999999], n]]]], {n, 2, 18}]]\)], "Input"], Cell[BoxData[ InterpretationBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\({2, 3, 11}\)}, {\({5, 7, 19}\)}, {\({2, 3, 1109}\)}, {\({3, 7, 3169}\)}, {\({3, 17, 13049}\)}, {\({5, 109, 12211}\)}, {\({2, 3, 29, 97, 3943}\)}, {\({3, 11, 13, 23, 67447}\)}, {\({3, 19, 116754307}\)}, {\({3, 11, 2016665303}\)}, {\({3, 53, 127, 179, 184117}\)}, {\({3, 7, 11, 63313, 455033}\)}, {\({3, 43, 515891124031}\)}, {\({3, 11, 13, 47, 33005978773}\)}, {\({3, 293, 773, 6791, 1442267}\)}, {\({3, 11, 61, 2238013, 14772071}\)}, {\({3, 2063, 107529415091291}\)} }], "\[NoBreak]", ")"}], MatrixForm[ {{2, 3, 11}, {5, 7, 19}, {2, 3, 1109}, {3, 7, 3169}, {3, 17, 13049}, {5, 109, 12211}, {2, 3, 29, 97, 3943}, {3, 11, 13, 23, 67447}, {3, 19, 116754307}, {3, 11, 2016665303}, {3, 53, 127, 179, 184117}, {3, 7, 11, 63313, 455033}, {3, 43, 515891124031}, {3, 11, 13, 47, 33005978773}, {3, 293, 773, 6791, 1442267}, {3, 11, 61, 2238013, 14772071}, {3, 2063, 107529415091291}}]]], "Output"] }, Open ]], Cell["\<\ Differences between the real and predicted density of \[Micro](n) = -1.\ \>", "Text"], Cell[BoxData[ \(\(ListPlot[ FoldList[Plus, 0, Table[If[MoebiusMu[n] \[Equal] \(-1\), 1, 0], {n, 1, 100000 - 1}]] - Table[n\ \((3/Pi^2)\), {n, 1, 100000}], PlotJoined \[Rule] True, AspectRatio \[Rule] 1/7];\)\)], "Input"], Cell["\<\ In the \"positive one\" bin, M\[ODoubleDot]bius put all the numbers that \ factor into an even number of distinct primes. For completeness, he put 1 \ into this bin.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(PositiveOneBin\ = \ Flatten[Position[Table[MoebiusMu[n], {n, 1, 100}], 1]]\)], "Input"], Cell[BoxData[ \({1, 6, 10, 14, 15, 21, 22, 26, 33, 34, 35, 38, 39, 46, 51, 55, 57, 58, 62, 65, 69, 74, 77, 82, 85, 86, 87, 91, 93, 94, 95}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(IntegerDigits[69999069090909090990909090] // Length\)], "Input"], Cell[BoxData[ \(26\)], "Output"] }, Open ]], Cell["\<\ I found 69999069090909090990909090 with the same method as above. This time, \ I restricted the search to numbers with only three digits. I believe this is \ the longest possible. No infinite \ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Table[ MoebiusMu[ FromDigits[Take[IntegerDigits[69999069090909090990909090], n]]], {n, 2, 26}]\)], "Input"], Cell[BoxData[ \({1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1, 1}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(MatrixForm[ Table[Map[First, FactorInteger[ FromDigits[ Take[IntegerDigits[69999069090909090990909090], n]]]], {n, 2, 26}]]\)], "Input"], Cell[BoxData[ InterpretationBox[ RowBox[{"(", "\[NoBreak]", GridBox[{ {\({3, 23}\)}, {\({3, 233}\)}, {\({3, 2333}\)}, {\({3, 23333}\)}, {\({2, 3, 5, 23333}\)}, {\({2, 3, 101, 11551}\)}, {\({3, 7, 29, 114941}\)}, {\({2, 3, 5, 7, 29, 114941}\)}, {\({3, 101, 149, 155047}\)}, {\({2, 3, 5, 101, 149, 155047}\)}, {\({3, 19, 102199, 120163}\)}, {\({2, 3, 5, 19, 102199, 120163}\)}, {\({3, 7, 101, 33002861429}\)}, {\({2, 3, 5, 7, 101, 33002861429}\)}, {\({3, 11, 79, 109, 25343, 972001}\)}, {\({2, 3, 5, 11, 79, 109, 25343, 972001}\)}, {\({3, 101, 173, 467, 1511, 18924403}\)}, {\({3, 11, 13, 16316799321890231}\)}, {\({2, 3, 5, 11, 13, 16316799321890231}\)}, {\({3, 7, 61, 101, 45497, 118915646137}\)}, {\({2, 3, 5, 7, 61, 101, 45497, 118915646137}\)}, {\({3, 72907, 178259, 1795354918031}\)}, {\({2, 3, 5, 72907, 178259, 1795354918031}\)}, {\({3, 13, 101, 1543, 5285989, 217878362453}\)}, {\({2, 3, 5, 13, 101, 1543, 5285989, 217878362453}\)} }], "\[NoBreak]", ")"}], MatrixForm[ {{3, 23}, {3, 233}, {3, 2333}, {3, 23333}, {2, 3, 5, 23333}, {2, 3, 101, 11551}, {3, 7, 29, 114941}, {2, 3, 5, 7, 29, 114941}, {3, 101, 149, 155047}, {2, 3, 5, 101, 149, 155047}, {3, 19, 102199, 120163}, {2, 3, 5, 19, 102199, 120163}, {3, 7, 101, 33002861429}, {2, 3, 5, 7, 101, 33002861429}, {3, 11, 79, 109, 25343, 972001}, {2, 3, 5, 11, 79, 109, 25343, 972001}, {3, 101, 173, 467, 1511, 18924403}, {3, 11, 13, 16316799321890231}, {2, 3, 5, 11, 13, 16316799321890231}, {3, 7, 61, 101, 45497, 118915646137}, {2, 3, 5, 7, 61, 101, 45497, 118915646137}, {3, 72907, 178259, 1795354918031}, {2, 3, 5, 72907, 178259, 1795354918031}, {3, 13, 101, 1543, 5285989, 217878362453}, {2, 3, 5, 13, 101, 1543, 5285989, 217878362453}}]]], "Output"] }, Open ]], Cell["For amazing factorability demonstrations, use PowerMod. ", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(PowerMod[10, 10^100, 316912650057057350374175801344000001]\)], "Input"], Cell[BoxData[ \(316912650057057350374175801344000000\)], "Output"] }, Open ]], Cell[TextData[{ "More Riemann Zeta function zeroes can be found at ", ButtonBox["Andrew Odlyzko: Tables of zeros of the Riemann zeta function", ButtonData:>{ URL[ "http://www.dtc.umn.edu/%7Eodlyzko/zeta_tables/"], None}, ButtonStyle->"Hyperlink"], "." }], "Text"], Cell[BoxData[ RowBox[{ RowBox[{\(\[Zeta]\_0\), " ", "=", StyleBox[\(1/2\ + \ I {{14.134725142}, {21.022039639}, {25.010857580}, \ {30.424876126}, {32.935061588}, {37.586178159}, {40.918719012}, \ {43.327073281}, {48.005150881}, {49.773832478}, {52.970321478}, \ {56.446247697}, {59.347044003}, {60.831778525}, {65.112544048}, \ {67.079810529}, {69.546401711}, {72.067157674}, {75.704690699}, \ {77.144840069}, {79.337375020}, {82.910380854}, {84.735492981}, \ {87.425274613}, {88.809111208}, {92.491899271}, {94.651344041}, \ {95.870634228}, {98.831194218}, {101.317851006}, {103.725538040}, \ {105.446623052}, {107.168611184}, {111.029535543}, {111.874659177}, \ {114.320220915}, {116.226680321}, {118.790782866}, {121.370125002}, \ {122.946829294}, {124.256818554}, {127.516683880}, {129.578704200}, \ {131.087688531}, {133.497737203}, {134.756509753}, {138.116042055}, \ {139.736208952}, {141.123707404}, {143.111845808}, {146.000982487}, \ {147.422765343}, {150.053520421}, {150.925257612}, {153.024693811}, \ {156.112909294}, {157.597591818}, {158.849988171}, {161.188964138}, \ {163.030709687}, {165.537069188}, {167.184439978}, {169.094515416}, \ {169.911976479}, {173.411536520}, {174.754191523}, {176.441434298}, \ {178.377407776}, {179.916484020}, {182.207078484}, {184.874467848}, \ {185.598783678}, {187.228922584}, {189.416158656}, {192.026656361}, \ {193.079726604}, {195.265396680}, {196.876481841}, {198.015309676}, \ {201.264751944}, {202.493594514}, {204.189671803}, {205.394697202}, \ {207.906258888}, {209.576509717}, {211.690862595}, {213.347919360}, \ {214.547044783}, {216.169538508}, {219.067596349}, {220.714918839}, \ {221.430705555}, {224.007000255}, {224.983324670}, {227.421444280}, \ {229.337413306}, {231.250188700}, {231.987235253}, {233.693404179}, \ {236.524229666}}\), FontSize->10, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}]}], StyleBox[";", FontSize->10, FontWeight->"Plain", FontSlant->"Plain", FontTracking->"Plain", FontVariations->{"Underline"->False, "Outline"->False, "Shadow"->False, "StrikeThrough"->False, "Masked"->False, "CompatibilityType"->0, "RotationAngle"->0}]}]], "Input"], Cell[TextData[{ "The ", Cell[BoxData[ \(\[Zeta]\_0\)]], " powers correlate nicely with MoebiusMu. I do not know if this is a new \ technique. To me, it was an obvious consequence of ", Cell[BoxData[ FormBox[ RowBox[{\(\[Sum]\+\(n = 1\)\%\[Infinity]\), FractionBox[ RowBox[{ TagBox["\[Mu]", MoebiusMu], "(", "n", ")"}], \(n\^s\)]}], TraditionalForm]]], "= ", Cell[BoxData[ FormBox[ FractionBox["1", TagBox[ RowBox[{"\[Zeta]", "(", TagBox["s", Zeta, Editable->True], ")"}], InterpretTemplate[ Function[ BoxForm`e$, Zeta[ BoxForm`e$]]]]], TraditionalForm]]], ". Note that it only has a ", StyleBox["tendancy", FontWeight->"Bold"], " to be correct." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Map[{First[#], Length[#]} &, Split[Sort[ Table[{MoebiusMu[ n], \(Sign[Re[n^\[Zeta]\_0[\([1]\)]]]\)[\([1]\)]}, {n, 1, 1000}]]]]\)], "Input"], Cell[BoxData[ \({{{\(-1\), \(-1\)}, 177}, {{\(-1\), 1}, 126}, {{0, \(-1\)}, 199}, {{0, 1}, 193}, {{1, \(-1\)}, 134}, {{1, 1}, 171}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Table[ If[MoebiusMu[n] \[Equal] 1, Sum[Re[n^\[Zeta]\_0[\([k]\)]], {k, 1, 100}], 0], {n, 1, 100}]\)], "Input"], Cell[BoxData[ \({{100}, 0, 0, 0, 0, {4.115365117136966`}, 0, 0, 0, {5.85559407696141`}, 0, 0, 0, {9.049951742081086`}, {12.307007046904623`}, 0, 0, 0, 0, 0, {9.98993870538377`}, {22.395904404615784`}, 0, 0, 0, {10.594726283448974`}, 0, 0, 0, 0, 0, 0, {4.5096950572998455`}, {15.040904091291923`}, {21.83920661754055`}, 0, 0, {34.409367808956304`}, {40.58707604182649`}, 0, 0, 0, 0, 0, 0, {34.486308211146984`}, 0, 0, 0, 0, {15.185729320720425`}, 0, 0, 0, {31.853812439706353`}, 0, {10.05841077482352`}, {67.84585524368478`}, 0, 0, 0, {60.864702904514054`}, 0, 0, {4.564378166683675`}, 0, 0, 0, {6.143321934104899`}, 0, 0, 0, 0, {45.78645295908575`}, 0, 0, {43.85602875063138`}, 0, 0, 0, 0, {3.83257949472692`}, 0, 0, {97.26853169135951`}, {\(-13.839903410400206`\)}, \ {99.1413666294979`}, 0, 0, 0, {63.15822952711776`}, 0, {46.30517875326013`}, {46.62285494152198`}, {57.14297929632877`}, 0, 0, 0, 0, 0}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Table[ If[MoebiusMu[n] \[Equal] \(-1\), Sum[Re[n^\[Zeta]\_0[\([k]\)]], {k, 1, 100}], 0], {n, 1, 100}]\)], "Input"], Cell[BoxData[ \({0, {\(-24.653438980990042`\)}, {\(-40.197805660378116`\)}, 0, {\(-59.27684964061668`\)}, 0, {\(-69.42450737675598`\)}, 0, 0, 0, {\(-86.46930900054858`\)}, 0, {\(-96.43397590855118`\)}, 0, 0, 0, {\(-107.96227604521359`\)}, 0, {\(-106.99220881457148`\)}, 0, 0, 0, {\(-107.48220754534081`\)}, 0, 0, 0, 0, 0, {\(-104.33650028305203`\)}, {\(-11.493397007302466`\)}, \ {\(-121.29049147308083`\)}, 0, 0, 0, 0, 0, {\(-103.73466862385295`\)}, 0, 0, 0, {\(-96.44716593372107`\)}, {59.46817694772623`}, \ {\(-106.82161106006306`\)}, 0, 0, 0, {\(-120.14395844251646`\)}, 0, 0, 0, 0, 0, {\(-124.57576328862032`\)}, 0, 0, 0, 0, 0, {\(-138.6127479091415`\)}, 0, {\(-132.0766633545637`\)}, 0, 0, 0, 0, {62.91603888094935`}, {\(-138.2208125843921`\)}, 0, 0, {60.31113664155061`}, {\(-137.3973119628577`\)}, 0, {\(-133.2701072122339`\)}, 0, 0, 0, 0, {28.241664083762952`}, {\(-116.41900851484498`\)}, 0, 0, 0, {\(-97.778321708839`\)}, 0, 0, 0, 0, 0, {\(-113.29488389752885`\)}, 0, 0, 0, 0, 0, 0, 0, {\(-124.1433296803938`\)}, 0, 0, 0}\)], "Output"] }, Open ]], Cell[BoxData[ \(\(ListPlot[ FoldList[Plus, 0, Flatten[Table[ Sign[Re[1/n^\[Zeta]\_0[\([1]\)]]]\ MoebiusMu[n], {n, 1, 100000}]]], PlotJoined \[Rule] True, AspectRatio \[Rule] 1/7];\)\)], "Input"], Cell[BoxData[ \(\(ListPlot[ FoldList[Plus, 0, Flatten[Table[ Sign[Re[1/n^\[Zeta]\_0[\([45]\)]]]\ MoebiusMu[n], {n, 1, 100000}]]], PlotJoined \[Rule] True, AspectRatio \[Rule] 1/7];\)\)], "Input"], Cell[BoxData[ \(\(ListPlot[ FoldList[Plus, 0, Flatten[Table[ Sign[Re[1/n^\[Zeta]\_0[\([92]\)]]]\ MoebiusMu[n], {n, 1, 100000}]]], PlotJoined \[Rule] True, AspectRatio \[Rule] 1/7];\)\)], "Input"], Cell[TextData[{ "For n = 1 to 20, \[Micro](n) = {1, -1, -1, 0, -1, 1, -1, 0, 0, 1, -1, 0, \ -1, 1, 1, 0, -1, 0, -1, 0}. The cumulative sum is thus {1, 0, -1, -1, -2, -1, \ -2, -2, -2, -1, -2, -2, -3, -2, -1, -1, -2, -2, -3, -3}. This is known as ", ButtonBox["Mertens Function", ButtonData:>{ URL[ "http://mathworld.wolfram.com/MertensFunction.html"], None}, ButtonStyle->"Hyperlink"], ", or M(x). In 1885, Stieltjes claimed to have a proof that |M(x)/", Cell[BoxData[ \(TraditionalForm\`\@x\)]], "| < 1 for all x. In 1897, Mertens said the same thing, and now gets all \ the credit. The first time |M(x)/", Cell[BoxData[ \(TraditionalForm\`\@x\)]], "| exceeds \.bd is at M(7725030629) = 43947." }], "Text"], Cell[BoxData[ \(\(ListPlot[ Drop[FoldList[Plus, 0, Table[MoebiusMu[n], {n, 1, 100000}]], 1]/ Sqrt[Range[100000]], PlotJoined \[Rule] True, AspectRatio \[Rule] 1/7, PlotRange \[Rule] {\(- .5\), .5}];\)\)], "Input"], Cell[BoxData[ \(\(ComplexCircle\ = \ Take[Sort[ Flatten[Table[a + b\ I, {a, \(-100\), 100}, {b, \(-100\), 100}]], Abs[#1] < Abs[#2] &], 10000];\)\)], "Input"], Cell[TextData[{ "The density of the squarefree Gaussian integers is ", Cell[BoxData[ FormBox[ FractionBox["6", RowBox[{\(\[Pi]\^2\), " ", TagBox["C", Function[ {}, Catalan]]}]], TraditionalForm]]], ", where C is the Catalan constant. " }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ FormBox[ RowBox[{"N", "[", RowBox[{ FractionBox["6", RowBox[{\(\[Pi]\^2\), " ", TagBox["C", Function[ {}, Catalan]]}]], ",", "50"}], "]"}], TraditionalForm]], "Input"], Cell[BoxData[ \(0.6637008046138534607214316576170486913583957967748083872376602414372`\ 50. \)], "Output"] }, Open ]], Cell["Here's a plot showing a convergence to that value.", "Text"], Cell[BoxData[ \(\(ListPlot[ FoldList[Plus, 0, Table[If[ Max[Map[Last, FactorInteger[ComplexCircle[\([n]\)]]]]\ \[Equal] \ 1, 1, 0], {n, 1, 10000}]]/Range[10001], PlotJoined \[Rule] True, AspectRatio \[Rule] \ 1/7];\)\)], "Input"], Cell["\<\ There is a pattern to how Re[1/n^(14.1347251417347 I+1/2)] changes sign.\ \>", "Text"], Cell[BoxData[ \(\(Plot[Re[1/n^\((14.1347251417347\ I + 1/2)\)], {n, 1, 2000}, AspectRatio \[Rule] \ 1/7, \ PlotRange \[Rule] \ {\(- .1\), .1}];\)\)], "Input"], Cell["First, we find the sign changes.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(signchanges\ = Flatten[Position[ Table[Sign[ Re[1\/n\^\(14.1347251417347\ \[ImaginaryI] + 1\/2\)] Re[1\/\((n + 1)\)\^\(14.1347251417347\ \[ImaginaryI] + \ 1\/2\)]], {n, 1, 2000}], \(-1\)]]\)], "Input"], Cell[BoxData[ \({1, 3, 4, 5, 6, 8, 10, 12, 16, 20, 25, 31, 39, 48, 61, 76, 95, 118, 148, 185, 231, 289, 361, 451, 563, 703, 879, 1097, 1371, 1712}\)], "Output"] }, Open ]], Cell["\<\ Next, we note that there is a ratio between successive terms.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(N[Drop[signchanges, 1]/Drop[signchanges, \(-1\)]]\)], "Input"], Cell[BoxData[ \({3.`, 1.3333333333333333`, 1.25`, 1.2`, 1.3333333333333333`, 1.25`, 1.2`, 1.3333333333333333`, 1.25`, 1.25`, 1.24`, 1.2580645161290323`, 1.2307692307692308`, 1.2708333333333333`, 1.2459016393442623`, 1.25`, 1.2421052631578948`, 1.2542372881355932`, 1.25`, 1.2486486486486486`, 1.251082251082251`, 1.2491349480968859`, 1.2493074792243768`, 1.2483370288248337`, 1.2486678507992894`, 1.2503556187766713`, 1.248009101251422`, 1.2497721057429352`, 1.24872355944566`}\)], "Output"] }, Open ]], Cell["Once the ratio is determined, a constant can be sought.", "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(signchanges/Table[1.2489^k, {k, 1, 30}]\)], "Input"], Cell[BoxData[ \({0.8007046200656578`, 1.9233836657834682`, 2.053416249802192`, 2.0552248476681405`, 1.9747536369619416`, 2.10825914747585`, 2.110116049599498`, 2.0274956037468153`, 2.1645667961105137`, 2.166473292608009`, 2.168381468300113`, 2.152928994068493`, 2.168727338604326`, 2.137243076533147`, 2.1747775987622773`, 2.169556390028905`, 2.1714672812363935`, 2.159653245895499`, 2.1688827211507746`, 2.170793019007501`, 2.1703561291373505`, 2.1741484760119447`, 2.174557485577792`, 2.175266979530539`, 2.1742864265577095`, 2.173882263809659`, 2.1764159685430036`, 2.1748634292982607`, 2.176382134468243`, 2.176074662236471`}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Table[Round[2.17607\ \ 1.2489^k], {k, 1, 30}]\ - \ signchanges\)], "Input"], Cell[BoxData[ \({2, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, 0, 1, 0, 0, 1, 0, 0, 0, 0, 1, 0, 1, 0, 0}\)], "Output"] }, Open ]], Cell[TextData[{ "Good enough. Now, we can use logarithms to tackle ", Cell[BoxData[ \(TraditionalForm\`10\^\(10\^100\)\)]], "+1. So far as I know, no-one has ever predicted the number of factors for \ a number this large before. Note that ", StyleBox["k", FontSlant->"Italic"], " represents the number of sign changes." }], "Text"], Cell[CellGroupData[{ Cell[BoxData[ \(Solve[2.17607\ \ 1.2489^k\ \[Equal] \ 10^x, k]\)], "Input"], Cell[BoxData[ RowBox[{\(Solve::"ifun"\), \(\(:\)\(\ \)\), "\<\"Inverse functions are \ being used by \\!\\(Solve\\), so some solutions may not be found; use Reduce \ for complete solution information. \ \\!\\(\\*ButtonBox[\\\"More\[Ellipsis]\\\", ButtonStyle->\\\"RefGuideLinkText\ \\\", ButtonFrame->None, ButtonData:>\\\"Solve::ifun\\\"]\\)\"\>"}]], \ "Message"], Cell[BoxData[ \({{k \[Rule] 4.499171084007631`\ Log[0.459544040403112`\ 10.`\^x]}}\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(Round[4.49917\ Log[0.459544]\ + \ 4.49917\ Log[10]\ 10^100]\)], "Input"], Cell[BoxData[ \(103597217728460212135830325440780130056164015486160138409326721976406801\ 160712681646102396432778526720\)], "Output"] }, Open ]], Cell[TextData[{ "We are interested in the number of sign changes. Out near a Googleplex, \ the sign has changed an even number of times, so there is a slight suggestion \ that googolplex+1 has an even number of factors. ", StyleBox["Much", FontWeight->"Bold"], " more accuracy would be desirable." }], "Text"], Cell["RasterGraphics code by Oyvind Tafjord of Wolfram Research.", "Text"], Cell[BoxData[ \(RasterGraphics[state_, colors_: 2, size_: 1] := With[{dim = Reverse[Dimensions[state]]}, Graphics[Raster[Reverse[1 - state/\((colors - 1)\)]], AspectRatio \[Rule] Automatic, PlotRange \[Rule] {{0, dim[\([1]\)]}, {0, dim[\([2]\)]}}, ImageSize \[Rule] size*dim + 1]]\)], "Input"], Cell[BoxData[ \(GaussianMu[a_]\ := \ With[{jj = FactorInteger[a, GaussianIntegers \[Rule] \ True]}, With[{kk = If[jj \[Equal] {}, {}, If[Abs[jj[\([1, 1]\)]] == 1, Drop[jj, 1], jj]]}, If[Max[Map[Last, kk]] > 1, 0, 2 Mod[Length[kk] + 1, 2] - 1]]]\)], "Input"], Cell["\<\ Nice pictures of the behavior of the Moebius function on Gaussian integers \ can be made.\ \>", "Text"], Cell[BoxData[ \(\(Show[ RasterGraphics[ Table[If[GaussianMu[a + b\ I] \[Equal] \(-1\), 1, 0], {a, \(-200\), 200}, {b, \(-200\), 200}]]];\)\)], "Input"], Cell[CellGroupData[{ Cell[BoxData[ \(Integrate[ArcTan[x]/x, {x, 0, 1}]\)], "Input"], Cell[BoxData[ \(Catalan\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(\((Zeta[2, 1/4]\ - \ Zeta[2, 3/4])\)/16\)], "Input"], Cell[BoxData[ \(Catalan\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(16/\((Zeta[2] \((Zeta[2, 1/4]\ - \ Zeta[2, 3/4])\))\) // FullSimplify\)], "Input"], Cell[BoxData[ \(6\/\(Catalan\ \[Pi]\^2\)\)], "Output"] }, Open ]], Cell[CellGroupData[{ Cell[BoxData[ \(FactorInteger[11 + 3 I]\)], "Input"], Cell[BoxData[ \({{\(-\[ImaginaryI]\), 1}, {1 + \[ImaginaryI], 1}, {2 + \[ImaginaryI], 1}, {3 + 2\ \[ImaginaryI], 1}}\)], "Output"] }, Open ]], Cell["\<\ Gauss observed that for numbers one less than a prime, MoebiusMu[p-1] is \ equal to the sum of the primitive roots of p, mod p.\ \>", "Text"], Cell[CellGroupData[{ Cell[BoxData[{ \(PrimitiveRootQ[a_Integer, p_Integer] := Block[{fac, res}, fac = FactorInteger[p - 1]; \[IndentingNewLine]res = Table[PowerMod[a, \((p - 1)\)/fac[\([i, 1]\)], p], {i, Length[fac]}]; \(! MemberQ[res, 1]\)]\n\), "\[IndentingNewLine]", \(PrimitiveRoots[p_Integer] := Select[Range[p - 1], PrimitiveRootQ[#, p] &]\n\), "\[IndentingNewLine]", \(Table[{Prime[n] - 1, MoebiusMu[Prime[n] - 1], Mod[Total[PrimitiveRoots[Prime[n]]], Prime[n], \(-1\)]}, {n, 2, 100}]\)}], "Input"], Cell[BoxData[ \({{2, \(-1\), \(-1\)}, {4, 0, 0}, {6, 1, 1}, {10, 1, 1}, {12, 0, 0}, {16, 0, 0}, {18, 0, 0}, {22, 1, 1}, {28, 0, 0}, {30, \(-1\), \(-1\)}, {36, 0, 0}, {40, 0, 0}, {42, \(-1\), \(-1\)}, {46, 1, 1}, {52, 0, 0}, {58, 1, 1}, {60, 0, 0}, {66, \(-1\), \(-1\)}, {70, \(-1\), \(-1\)}, {72, 0, 0}, {78, \(-1\), \(-1\)}, {82, 1, 1}, {88, 0, 0}, {96, 0, 0}, {100, 0, 0}, {102, \(-1\), \(-1\)}, {106, 1, 1}, {108, 0, 0}, {112, 0, 0}, {126, 0, 0}, {130, \(-1\), \(-1\)}, {136, 0, 0}, {138, \(-1\), \(-1\)}, {148, 0, 0}, {150, 0, 0}, {156, 0, 0}, {162, 0, 0}, {166, 1, 1}, {172, 0, 0}, {178, 1, 1}, {180, 0, 0}, {190, \(-1\), \(-1\)}, {192, 0, 0}, {196, 0, 0}, {198, 0, 0}, {210, 1, 1}, {222, \(-1\), \(-1\)}, {226, 1, 1}, {228, 0, 0}, {232, 0, 0}, {238, \(-1\), \(-1\)}, {240, 0, 0}, {250, 0, 0}, {256, 0, 0}, {262, 1, 1}, {268, 0, 0}, {270, 0, 0}, {276, 0, 0}, {280, 0, 0}, {282, \(-1\), \(-1\)}, {292, 0, 0}, {306, 0, 0}, {310, \(-1\), \(-1\)}, {312, 0, 0}, {316, 0, 0}, {330, 1, 1}, {336, 0, 0}, {346, 1, 1}, {348, 0, 0}, {352, 0, 0}, {358, 1, 1}, {366, \(-1\), \(-1\)}, {372, 0, 0}, {378, 0, 0}, {382, 1, 1}, {388, 0, 0}, {396, 0, 0}, {400, 0, 0}, {408, 0, 0}, {418, \(-1\), \(-1\)}, {420, 0, 0}, {430, \(-1\), \(-1\)}, {432, 0, 0}, {438, \(-1\), \(-1\)}, {442, \(-1\), \(-1\)}, {448, 0, 0}, {456, 0, 0}, {460, 0, 0}, {462, 1, 1}, {466, 1, 1}, {478, 1, 1}, {486, 0, 0}, {490, 0, 0}, {498, \(-1\), \(-1\)}, {502, 1, 1}, {508, 0, 0}, {520, 0, 0}, {522, 0, 0}, {540, 0, 0}}\)], "Output"] }, Open ]] }, Open ]] }, Open ]] }, FrontEndVersion->"5.0 for Microsoft Windows", ScreenRectangle->{{0, 1024}, {0, 723}}, WindowSize->{883, 524}, WindowMargins->{{11, Automatic}, {Automatic, 0}} ] (******************************************************************* Cached data follows. 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